Theorem elelpwi 3733

Description: If A belongs to a part of C then A belongs to C. (Contributed by FL, 3-Aug-2009.)

Assertion

Ref	Expression
elelpwi	|- ((A e. B /\ B e. ~PC) -> A e. C)

Proof of Theorem elelpwi

Step	Hyp	Ref	Expression
1		elpwi 3731	. . 3 |- (B e. ~PC -> B C_ C)
2	1	sseld 3273	. 2 |- (B e. ~PC -> (A e. B -> A e. C))
3	2	impcom 419	1 |- ((A e. B /\ B e. ~PC) -> A e. C)

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1710  ~Pcpw 3723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pw 3725

This theorem is referenced by: (None)